Furthermore, their main result establishes the stability property: under the same degree assumption, if there is no perfect matching then there must be a space or divisibility barrier. This allows the use of the stability method in proving exact results.
Besides recovering previous results, the authors apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemeredi Theorem.
Here they prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical they defer it to a subsequent paper.
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